"quadratic residue" meaning in All languages combined

{
  "antonyms": [
    {
      "sense": "antonym(s) of “integer expressible as a square modulo n”",
      "word": "quadratic nonresidue"
    }
  ],
  "forms": [
    {
      "form": "quadratic residues",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "quadratic residue (plural quadratic residues)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Entries with translation boxes",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with Danish translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with Italian translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Number theory",
          "orig": "en:Number theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 52:",
          "text": "There are many small tables of quadratic residues giving for the first few primes p the positive quadratic residues of p arranged in order of their size.",
          "type": "quote"
        },
        {
          "text": "1991, John Stillwell (translator), Peter Gustav Lejeune Dirichlet, Lectures on Number Theory, [1863, Vorlesungen über Zahlentheorie], American Mathematical Society, London Mathematical Society, page 82,\nIf we now make the assumption that q is a quadratic residue of all odd primes z not greater than 2m + 1, then it follows from earlier theorems (§37) that the prime q, since it is ≡ 1 (mod 8) and hence a quadratic residue of each power of 2, is also a quadratic residue of each number which has no odd prime factors except the prime numbers z"
        },
        {
          "text": "1999, Dinakar Ramakrishnan, Robert J. Valenza, Tsinghua University Press [清华大学出版社有限公司], Fourier Analysis on Number Fields, page 213,\nOf special importance here is the quadratic reciprocity law, which for primes p and q gives a precise relationship between the status of p as a quadratic residue mod q and the status of q as a quadratic residue mod p."
        }
      ],
      "glosses": [
        "For given positive integer n, any integer that is congruent to some square m² modulo n."
      ],
      "id": "en-quadratic_residue-en-noun-B5dFNw-Y",
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "congruent",
          "congruent"
        ],
        [
          "square",
          "square"
        ],
        [
          "modulo",
          "modulo"
        ]
      ],
      "qualifier": "modular arithmetic",
      "raw_glosses": [
        "(number theory, modular arithmetic) For given positive integer n, any integer that is congruent to some square m² modulo n."
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "translations": [
        {
          "code": "da",
          "lang": "Danish",
          "sense": "integer that is congruent to the square of some other number, to a given modulus",
          "tags": [
            "common-gender"
          ],
          "word": "kvadratisk rest"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "integer that is congruent to the square of some other number, to a given modulus",
          "tags": [
            "masculine"
          ],
          "word": "residuo quadratico"
        }
      ],
      "wikipedia": [
        "quadratic residue"
      ]
    }
  ],
  "word": "quadratic residue"
}

{
  "antonyms": [
    {
      "sense": "antonym(s) of “integer expressible as a square modulo n”",
      "word": "quadratic nonresidue"
    }
  ],
  "forms": [
    {
      "form": "quadratic residues",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "quadratic residue (plural quadratic residues)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Terms with Danish translations",
        "Terms with Italian translations",
        "en:Number theory"
      ],
      "examples": [
        {
          "ref": "1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 52:",
          "text": "There are many small tables of quadratic residues giving for the first few primes p the positive quadratic residues of p arranged in order of their size.",
          "type": "quote"
        },
        {
          "text": "1991, John Stillwell (translator), Peter Gustav Lejeune Dirichlet, Lectures on Number Theory, [1863, Vorlesungen über Zahlentheorie], American Mathematical Society, London Mathematical Society, page 82,\nIf we now make the assumption that q is a quadratic residue of all odd primes z not greater than 2m + 1, then it follows from earlier theorems (§37) that the prime q, since it is ≡ 1 (mod 8) and hence a quadratic residue of each power of 2, is also a quadratic residue of each number which has no odd prime factors except the prime numbers z"
        },
        {
          "text": "1999, Dinakar Ramakrishnan, Robert J. Valenza, Tsinghua University Press [清华大学出版社有限公司], Fourier Analysis on Number Fields, page 213,\nOf special importance here is the quadratic reciprocity law, which for primes p and q gives a precise relationship between the status of p as a quadratic residue mod q and the status of q as a quadratic residue mod p."
        }
      ],
      "glosses": [
        "For given positive integer n, any integer that is congruent to some square m² modulo n."
      ],
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "congruent",
          "congruent"
        ],
        [
          "square",
          "square"
        ],
        [
          "modulo",
          "modulo"
        ]
      ],
      "qualifier": "modular arithmetic",
      "raw_glosses": [
        "(number theory, modular arithmetic) For given positive integer n, any integer that is congruent to some square m² modulo n."
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "quadratic residue"
      ]
    }
  ],
  "translations": [
    {
      "code": "da",
      "lang": "Danish",
      "sense": "integer that is congruent to the square of some other number, to a given modulus",
      "tags": [
        "common-gender"
      ],
      "word": "kvadratisk rest"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "integer that is congruent to the square of some other number, to a given modulus",
      "tags": [
        "masculine"
      ],
      "word": "residuo quadratico"
    }
  ],
  "word": "quadratic residue"
}